CONCERNING A CONJECTURE OF MARSHALL HALL

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1 CONCERNING A CONJECTURE OF MARSHALL HALL RICHARD SINKHORN Introdction. An «X«matrix A is said to be dobly stochastic if Oij; = and if ï i aik = jj_i akj = 1 for all i and j. The set of «X«dobly stochastic matrices is denoted by fi. The permanent of an «X«matrix A is defined by n per A = II «*»«) a i-1 where the sm is taken over all permtations a of 1,, «. An nresolved conjectre of B. L. Van der Waerden states that the minimal vale of the permanent of the «X«dobly stochastic matrices is «!/«" and is niqely achieved at the matrix Jn in which every element is 1/«. Marshall Hall has made the following observation. Let A denote the collection of «X«(, l)-matrices (i.e. those for which every element is either or 1) which have exactly three ones in each row and each colmn. If A QAn, then \A QQ.n, and ths if the Van der Waerden conjectre is correct, per \A^n\/nn, i.e. per A _3"w!/«n. Ths, in particlar and therefore Inf per A = 3n«!/«n, AEA (1) lim ( Inf per A 1= + <*. n->» \A6An / Marshall Hall conjectres therefore that (1) holds. It is the prpose of this paper to show that the Hall conjectre is correct independent of the correctness of the van der Waerden conjectre. The following notations and definitions will be sed. An «X«matrix A is said to be partly decomposable if there exist permtation matrices P and Q sch that PAQ has the form where B and D are sqare. If no sch P and Q exist, the matrix is said to be flly indecomposable. If A is flly indecomposable, bt is sch that if any single a.y^o in A is replaced by the reslting Presented to the Society, Agst 3, 1968; received by the editors Janary 19,

2 198 RICHARD SINKHORN [April matrix becomes partly decomposable, then A is said to be nearly decomposable. An important theorem concerning nearly decomposable matrices follows. A proof may be fond in [2]. Theorem 1. Let A be a nonnegative nxn nearly decomposable (, 1)- matrix where «> 1. Then there exist permtation matrices P and Q and an integer s>\ sch that (2) PAQ = \Ai E2 A2 E3 A, (i Fi) 7,_i A,-i E, A,] where each Ei has exactly one entry eqal to 1 and each Ai is nearly decomposable. A set of elements aiff(i),, an<,(n) in an «X«matrix.4, where a isa permtation of 1,, w, is called a diagonal of A. Reslts and conseqences. Lemma 1. Let be a positive integer and sppose that Vi S; v2 ^ ^d^3. Then n v>< - «- e v* +1 ^ o. Proof. The reslt holds for = 1. If it holds for, then JI vt ( + i) Ev* + i = z'«+i IT "*-i-e»* ni=l > t=l *=1,1+ii«+ E»* i) +1 E * completing the indction. = (»+i 1)«+ (»«+1 1) E ^* 2î>«+i Jr=l ^ 2«+ 2z) 2z> +i è 2«2 >, Lemma 2. Leí w,, area q be integers where ^i and O^p^q, and sppose that Vi^v-i^ ^v^3. Then

3 1969] CONCERNING A CONJECTURE OF MARSHALL HALL 199 and 1+2»3«-» n»* = «+ 3? + B*. i=l Proof. Pt /(x) = 2*n"-ií'*-M-3*- E"-i»*+l- BY Lemma 1, /(o) = n»*-«-»*+i^o i-1 k~l /(i) = 2n»*-«-3-E»*+i= /(o)+( n»*- 3 ) = o. Jb=l /fc-1 \ fc=l / If arel, /'(*) = 2*( n^jln 2-3 è 2-3 In 2-3 = 3(ln 4-1) >. Ths/(x)^/(l)= if x=l. Then since 2*3'-* = 2", 2*3»*- «- 3? - Z % + 1 = /(?) =. t-i *=i Lemma 3. Let A be an nxn nearly decomposable (, I)-matrix with at least two and not more than three ones in each row. If at least one row of A contains three ones, then per A = 3. Proof. It follows from the Frobenis-König Theorem [l, pp ] that every positive element in a nonnegative flly indecomposable matrix lies on a positive diagonal. Assme A to be in the form (2). Then some A, is at least 2X2 and therefore has at least two positive diagonals. Whence per ^4,^2. Allowing for at least one positive diagonal of A to pass throgh the P<, we see that per A = 1 + JX*-i Per <=l+2=3. Lemma 4. 7/4«a flly indecomposable (, \)-matrix by 1, the permanent is increased. a zero is replaced Proof. If A is a flly indecomposable (, l)-matrix and if B is obtained from A by replacing a zero by 1, then B is still flly indecomposable. By the Frobenis-König Theorem every 1 in B, inclding the 1 not in A, lies on a positive diagonal. Ths B has at least one more positive diagonal than does A. Hence per P>per A. Theorem 2. Let A be a nearly decomposable nxn (, \)-matrix with m rows containing exactly three ones and n m rows containing exactly two ones. Then per.4 _m.

4 2 RICHARD SINKHORN [April Proof. The proof is by indction on re. We mst have re^2, and if» = 2, then m = Q and the reslt holds. In fact, if m =, the reslt holds for all «^ 2 ; whence if «> 2 we sppose that m >. We may assme that A is in the form (2). Since m>, at least one of the ^4<'s mst be larger than 1X1. Sppose that Ailt, Air are at least 2X2 and that each of Aiy, Aip, p^r, has exactly two ones in each row (and ths in each colmn) while Aip+1,, Air have mp+i,, mt rows with three ones, respectively. Note that m = r + Zt-p+i m*' Let mk- 2 for k=p + \,, t, and mk^3 for k = t + l,, r. Then by Lemma 3 and the indction hypothesis, per Ailt^3 for k=p + \,, t and per Ai}>.mk for k = t + l,, r. Since per Aik = 2 for k = \,, p, we have, allowing for at least one positive diagonal throgh the Ei, per A è 1 + Per ^,H r + 2"3<-p Ü mk. *=1 i-t+l The proof will be complete if we show that l+2p3,_i,hi_,+1 mk^r + Et-p+i w*- If í =, then p =, and the reslt is an immediate conseqence of Lemma 1. Hence sppose >. If t = r, the ineqality becomes 1+2P3,_P^/+ Et-p+i mk- We prove this by showing that l+2r3t-r^3t-2p. This is clear for p =. If <p^t, we have 1 + 2*3'-* ^ 1 + 2' > 3/ - 2 è 3/ - 2p. Then since mk^2 ii k = p+l,, t, Ths t E m ^ 2(t - p). t-p+i 1 + 2"3'-" ^t + 2(t-p)^t+ E rnkk=p+l Whence we can assme that r>t. Then by Lemma 2, 1 + 2*3'-' fl mk^(r-t) + 3t+ mk = r »* t í+i *=í+i *=<+i Bt EUï>+i mk^2(t-p)s2t, ii t>p, so r+2i+e*-i+i *»*» + Et-2>+i ^fc- and the reslt follows. Theorem 3. 7/^4 is a flly indecomposable member of A then per A'ïtn. r

5 1969] CONCERNING A CONJECTURE OF MARSHALL HALL 21 Proof. If a certain set of ones in A, k in nmber, are replaced by zeros, there reslts a nearly decomposable matrix A'. The matrix A', being flly indecomposable, has at least two ones in every row (and colmn), and, of corse, no more than three. Ths the k ones were removed from k different rows (and colmns). A' has k rows with exactly two ones and «k rows with exactly three ones. By Theorem 2, per A'^.n k. Bt by Lemma 4, A has at least k more positive diagonals than did A' (one at least for each of the k ones removed). Ths per A _ k + («k) = «. Theorem 4. Inf ea per A =«. Proof. Sppose AQA is partly decomposable. By permting rows and colmns we may sppose that A,-(A' Y \B Aj where Ai is rxr and Ai is («r)x(n r). Then since the row sms of Aa are all 3, the sm of the elements in ^4i is 3r. This is precisely the sm of the elements in Ai and P. Ths P= and in fact A = Ai Ai, where AiQAr and AiQAn-T- Clearly the process can be contined so that PAoQ = Ai Ak for some permtation matrices P and Q, where the Aj are flly indecomposable matrices in A y, j 1,, k. Hence, by Theorem 3, per Aj^nj. Now each row of A has three ones and therefore n^3,,7 = 1,, k, and we have k k k per A = IX per Aj è II wj = E ni = M> y-i i=i y-i where eqality can occr only if k = 1. The Marshall Hall conjectre stated in (1) is now immediate. References 1. Marvin Marcs and Henryk Mine, A srvey of matrix theory and matrix ineqalities, Allyn and Bacon, Boston, Richard Sinkhorn and Pal Knopp, Problems involving diagonal prodcts in nonnegative matrices, Trans. Amer. Math. Soc. 136 (1969), University of Hoston